| L(s) = 1 | − 3·3-s + 6·5-s + 7·7-s + 9·9-s − 4·11-s − 46·13-s − 18·15-s − 82·17-s + 84·19-s − 21·21-s + 44·23-s − 89·25-s − 27·27-s + 70·29-s − 152·31-s + 12·33-s + 42·35-s − 146·37-s + 138·39-s + 94·41-s + 488·43-s + 54·45-s − 32·47-s + 49·49-s + 246·51-s − 562·53-s − 24·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.536·5-s + 0.377·7-s + 1/3·9-s − 0.109·11-s − 0.981·13-s − 0.309·15-s − 1.16·17-s + 1.01·19-s − 0.218·21-s + 0.398·23-s − 0.711·25-s − 0.192·27-s + 0.448·29-s − 0.880·31-s + 0.0633·33-s + 0.202·35-s − 0.648·37-s + 0.566·39-s + 0.358·41-s + 1.73·43-s + 0.178·45-s − 0.0993·47-s + 1/7·49-s + 0.675·51-s − 1.45·53-s − 0.0588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 46 T + p^{3} T^{2} \) |
| 17 | \( 1 + 82 T + p^{3} T^{2} \) |
| 19 | \( 1 - 84 T + p^{3} T^{2} \) |
| 23 | \( 1 - 44 T + p^{3} T^{2} \) |
| 29 | \( 1 - 70 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 94 T + p^{3} T^{2} \) |
| 43 | \( 1 - 488 T + p^{3} T^{2} \) |
| 47 | \( 1 + 32 T + p^{3} T^{2} \) |
| 53 | \( 1 + 562 T + p^{3} T^{2} \) |
| 59 | \( 1 + 476 T + p^{3} T^{2} \) |
| 61 | \( 1 - 34 T + p^{3} T^{2} \) |
| 67 | \( 1 + 520 T + p^{3} T^{2} \) |
| 71 | \( 1 + 36 T + p^{3} T^{2} \) |
| 73 | \( 1 + 654 T + p^{3} T^{2} \) |
| 79 | \( 1 + 608 T + p^{3} T^{2} \) |
| 83 | \( 1 - 284 T + p^{3} T^{2} \) |
| 89 | \( 1 + 954 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1694 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.666392248814385207736220619717, −9.028743893145886793670695824249, −7.74610050318308810083414773910, −7.02601708607266513181583887659, −5.96431310708649090489380303177, −5.15615996374709278538446008846, −4.28684166920927863164172280389, −2.72958896321059291980123821313, −1.55394252125910904076068027458, 0,
1.55394252125910904076068027458, 2.72958896321059291980123821313, 4.28684166920927863164172280389, 5.15615996374709278538446008846, 5.96431310708649090489380303177, 7.02601708607266513181583887659, 7.74610050318308810083414773910, 9.028743893145886793670695824249, 9.666392248814385207736220619717
